solve 2sin^2(x)+3cos(x)=3

Question

the smallest non-negative solution, and the next smallest non-negative solution

Answer 1

Question: solve 2sin^2(x)+3cos(x)=3.

Thius is actually a quadratic equation, in disguise. The equation is,

2sin^2(x) + 3cos(x) = 3

2(1 - cos^2(x)) + 3cos(x) = 3

now let u = cos(x). The equation becomes,

2(1 - u^2) + 3u = 0

2 - 2u^2 + 3u =0

2u^2 - 3u - 2= 0

(2u + 1)( u - 2) = 0

u = -1/2, u = 2

cos(x) = -1/2, cos(x) = 2

ignore the cos(x) = 2 solution. It is a solution to the quadratic equation, but not to the trig equation, since |cos(x)| <=1.

cos(x) = -1/2

x = 2.pi/3 + 2n.pi